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Convergence Solutions

The document discusses various tests for determining the convergence or divergence of series, including the Harmonic Series, Integral Test, Root Test, Ratio Test, and Alternating Series Test. Key findings include that the harmonic series diverges, series of the form ∑ (1/n^p) converge for p > 1, and the geometric series converges if |r| < 1. Additionally, it covers the Limit Comparison Test, convergence of ∑ (n^2 / 2^n), and the Divergence Test.

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15 views2 pages

Convergence Solutions

The document discusses various tests for determining the convergence or divergence of series, including the Harmonic Series, Integral Test, Root Test, Ratio Test, and Alternating Series Test. Key findings include that the harmonic series diverges, series of the form ∑ (1/n^p) converge for p > 1, and the geometric series converges if |r| < 1. Additionally, it covers the Limit Comparison Test, convergence of ∑ (n^2 / 2^n), and the Divergence Test.

Uploaded by

talhafrazmuhammad
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Solutions to Convergence and Divergence Problems

1. Harmonic Series Convergence


The harmonic series is:
∑ (1/n)
To determine convergence, we use the Integral Test. Consider the improper integral:
∫ (1/x) dx from 1 to ∞ = ln(x) |₁^∞ = ∞
Since the integral diverges, the harmonic series also diverges.

2. Integral Test for ∑ (1/n^p)


Using the Integral Test, evaluate:
∫ (1/x^p) dx from 1 to ∞
For p ≠ 1, this evaluates as:
(x^(1-p))/(1-p) |₁^∞
If p > 1, the integral converges; otherwise, it diverges.
Thus, the series converges for p > 1 and diverges for p ≤ 1.

3. Root Test for ∑ (n / (2n+1)^n)


Using the Root Test:
limsup (n → ∞) (n/(2n+1)^n)^(1/n) = 0
Since the limit is less than 1, the series converges.

4. Ratio Test for ∑ (n! / 2^n)


Applying the Ratio Test:
lim (n → ∞) |(n+1)! / 2^(n+1) ÷ (n! / 2^n)|
= lim (n → ∞) (n+1) * (1/2) = ∞
Since the limit is greater than 1, the series diverges.

5. Alternating Harmonic Series


Using the Alternating Series Test:
- The sequence a_n = 1/n is decreasing.
- lim (n → ∞) a_n = 0.
Since both conditions hold, the series converges.

6. Geometric Series ∑ ar^n


A geometric series converges if |r| < 1.
For ∑ 3(0.5)^n:
- a = 3, r = 0.5
Since |0.5| < 1, the series converges.
7. Limit Comparison Test for ∑ ((5n+3) / (n^2+4))
Compare with ∑ (1/n), which diverges.
lim (n → ∞) ((5n+3) / (n^2+4)) ÷ (1/n)
= lim (n → ∞) (5n^2 + 3n) / (n^2 + 4) = 5
Since the limit is finite and nonzero, the series diverges.

8. Convergence of ∑ (n^2 / 2^n)


Using the Ratio Test:
lim (n → ∞) |(n+1)^2 / 2^(n+1) ÷ (n^2 / 2^n)|
= lim (n → ∞) ((n+1)^2 / n^2) * (1/2)
= (1 + 2/n + 1/n^2) * (1/2) = 1/2
Since the limit is less than 1, the series converges.

9. Integral Test for ∑ (1 / (n ln(n))^2)


Using the Integral Test:
∫ (1 / (x ln(x))^2) dx from 2 to ∞
Let u = ln(x), then du = dx/x.
∫ du / u^2 = -1/u | ln(2) to ∞
Since the integral converges, the series converges.

10. Divergence Test for ∑ (n / (n+1))


If lim (n → ∞) a_n ≠ 0, the series diverges.
lim (n → ∞) (n / (n+1)) = 1 ≠ 0.
Thus, the series diverges by the Divergence Test.

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